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Extrapolation of Charge Density: Ver. 1.0

Taisuke Ozaki, RCIS, JAIST

Let us consider an extrapolation $\bar{{\bf x}}_{n+1}$ of the coordinate at the $(n+1)$th molecular dynamic or geometry optimization step [1,2] by a linear combination of the previous coordinates $\{{\bf x}_{m}\}$ as
    $\displaystyle \bar{{\bf x}}_{n+1} = \sum_{m=n-(p-1)}^{n}\alpha_m {\bf x}_{m},$ (1)

where for $p$ three is an optimum choice in many cases. To fit well the coordinate $\bar{{\bf x}}_{n+1}$ to the real coordinate ${\bf x}_{n+1}$, we consider the minimization of a function $F$:
    $\displaystyle F = \vert \bar{{\bf x}}_{n+1}-{\bf x}_{n+1} \vert^2
- \lambda \left(1-\sum_{m=n-(p-1)}^{n} \alpha_m \right)$ (2)

with respect to $\{\alpha_i\}$ and $\lambda$. The conditions $\frac{\partial F}{\partial \alpha_m}=0$ and $\frac{\partial F}{\partial \lambda}=0$ leads to
    $\displaystyle \left(
\begin{array}{cccc}
\langle {\bf x}_{(n-(p-1))} \vert {\bf...
...\right)
=
\left(
\begin{array}{c}
0\\
0\\
\cdot\\
1
\end{array}\right).\quad$ (3)

By solving the linear equation, we may have an optimum choice of a set of $\{\alpha_m\}$. Then, it is assumed that the difference charge density $\Delta \rho_{i}^{\rm (out)}$ can be extrapolated well by the same set of coefficients $\{\alpha_m\}$ as
    $\displaystyle \rho_{n+1}^{\rm (in)} = \rho_{n+1}^{\rm (atom)}
+ \sum_{m=n-(p-1)}^{n}\alpha_m \Delta \rho_{m}^{\rm (out)},$ (4)

where $\rho_{n+1}^{\rm (atom)}$ is given by the superposition of atomic charge densities at ${\bf x}_{n+1}$. Using Eq. (4) it can be possible to estimate a good input charge density at the $(n+1)$th step in molecular dynamic simulations or geometry optimizations.

Bibliography

1
T. A. Arias, M. C. Payne, and J. D. Joannopoulos, Phys. Rev. B 45, 1538 (1992).

2
D. Alfe, Comp. Phys. Commun. 118, 32 (1999).

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2007-08-16